Wednesday, January 16, 2019 - 11:00
Humboldt-Universität, Institut für Mathematik
Rudower Chaussee 25, 12489 Berlin, Raum 2.006, Haus 2
Forschungsseminar "Algebra und Zahlentheorie"
Prof. E. Große-Klönne
Let $L$ be a local field with residue field of characteristic $p>0$ and let $G_L$ be its absolute Galois group. For $L=\mathbb{Q}_p$, a well-known theorem due to Fontaine says that there is an equivalence between the category of continuous finitely generated representations of $G_{\mathbb{Q}_p}$ over $\mathbb{Z}_p$ (resp. $\mathbb{F}_p$) and the category of \'{e}tale $(\varphi,\Gamma)$-modules over $A_{\mathbb{Q}_p}$ (resp. $\mathbb{F}_{p}((t))$). In a past seminar, a generalization for an arbitrary $L$ has been also discussed. In this talk, we look at another generalization due to Z\'{a}br\'{a}di. We introduce $(\varphi, \Gamma)$-modules over $\mathbb{F}_p[[t_1,\ldots, t_n]][\prod^{n}_{i=1}t_i^{-1}]$ and discuss the equivalence between the category of continuous representations of the $d$th direct power of $G_{\mathbb{Q}_p}$ on finite dimensional $\mathbb{F}_p$-vector spaces and the category of \'{e}tale $(\varphi,\Gamma)$-modules over this $d$-variable Laurent power series ring.
submitted by S. Schmidt (sschmidt@math.hu-berlin.de, 2093 1820)